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Chapter 8: Problem 26

The Bragg angle corresponding to a reflection for which \(\left(h^{2}+k^{2}+l^{2}\right)=8\) is found to be \(14.35^{\circ}\). Determine the lattice parameter of the crystal. X-rays or wavelength \(0.71 \AA\) are used. If there are two other reflections with smaller Bragg angles, what is the crystal structure?

Short Answer

Expert verified
The lattice parameter \( a \) of the crystal is approximately 4.068 Å. The plausible crystal structure is BCC.

Step by step solution

01

Understand Bragg's Law

Bragg's Law is given by the formula: \[ n\lambda = 2d \sin \theta \]where \( n \) is the order of reflection (usually 1 for first-order), \( \lambda \) is the wavelength of X-rays, \( d \) is the interplanar spacing, and \( \theta \) is the Bragg angle.
02

Substitute Given Values

For the first reflection, the Bragg angle \( \theta \) is given as \(14.35^{\circ}\), and the wavelength \( \lambda \) is 0.71 \( \AA \). Substitute these values into Bragg's Law:\[ 1 \(0.71 \AA \) = 2d \sin(14.35^{\circ}) \]Solve for \(d\).
03

Calculate \( d \)

Using the sine value approximation, \( \sin(14.35^{\circ}) \approx 0.247 \):\[ 0.71 = 2d \times 0.247 \]\[ d = \frac{0.71}{2 \times 0.247} \approx 1.437 \( \AA \) \]
04

Relate \( d \) to Lattice Parameter \( a \)

For a cubic crystal system, the interplanar spacing \( d \) is related to the lattice parameter \( a \) and the Miller indices \( (h, k, l) \) using the formula:\[ d = \frac{a}{\sqrt{h^{2} + k^{2} + l^{2}}} \]Given that \( h^{2} + k^{2} + l^{2} = 8 \), substitute \( d \approx 1.437 \AA \):\[ 1.437 = \frac{a}{\sqrt{8}} = \frac{a}{2\sqrt{2}} \]
05

Solve for Lattice Parameter \( a \)

Isolate \( a \):\[ a = 1.437 \times 2 \sqrt{2} \ \ \approx 4.068 \AA \]
06

Determine Crystal Structure

If there are two other reflections with smaller Bragg angles, it implies multiple sets of smaller Miller indices \((h, k, l)\) corresponding to reflections. Common cubic structures to have such reflections include FCC and BCC. With the given value and reflection conditions, the most probable crystal structure is BCC.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bragg angle
The Bragg angle, represented by the symbol \( \theta \), is a crucial parameter in the analysis of crystal structures using X-ray diffraction. It is the angle at which diffracted X-rays constructively interfere due to periodic lattice planes. This constructive interference occurs when the path difference between successive crystal planes is an integer multiple of the X-ray wavelength.
lattice parameter
The lattice parameter, often denoted by \( \text{a} \), is a physical dimension representing the unit cell size in a crystal lattice. This parameter is directly related to the arrangement and spacing of atoms in the crystal. For cubic systems, the relationship is simpler, as it involves the interplanar spacing d and the Miller indices. The exact value of the lattice parameter helps determine the type of crystal structure and its properties.
crystal structure determination
Crystal structure determination involves identifying the arrangement of atoms within a crystalline material. Using X-ray diffraction data, such as the Bragg angle and interplanar spacing, one can deduce the lattice parameter. This process relies heavily on Bragg's Law and the relationship between the lattice parameter and the Miller indices, enabling scientists to classify the crystal into specific types, like Body-Centered Cubic (BCC) or Face-Centered Cubic (FCC) structures.
X-ray diffraction
X-ray diffraction (XRD) is a powerful technique used to investigate the atomic structure of crystalline materials. By measuring the angles and intensities of diffracted X-rays, XRD can reveal details about the spacing between planes of atoms, the lattice parameter, and the overall crystal structure. This method is based on Bragg’s Law and is crucial for material science and chemistry.
Miller indices
Miller indices are a set of three numbers (h, k, l) that describe the orientation of a crystal plane in a lattice. These indices are fundamental in X-ray diffraction studies as they help relate the interplanar spacing and lattice parameter. For cubic systems, a common formula relating the indices is \( \left( h^2 + k^2 + l^2 \right) \). In the given exercise, knowing these values allowed for the calculation of the lattice parameter from the known Bragg angle.

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Most popular questions from this chapter

The mass absorption coefficient of aluminium for X-rays of a certain energy is \(0.027 \mathrm{~m}^{2} 7 \mathrm{~kg}\). Calculate the half-value thickness of aluminium for these \(X\) -rays. What thickness of aluminium would attenuate the X-rays by \(80 \%\) ? Density of aluminium is \(2700 \mathrm{~kg} / \mathrm{m}^{3}\).

The diamond structure is formed by the combination of two interpenetrating fce sub-lattices: the basis being \((000)\) and \((1 / 41 / 41 / 4)\). Find the structure factor of the basis and prove that if all indices are even the structure factor of the basis van?shes unless \(h+k+l=4 n\), where \(n\) is an integer.

Sylvine (KCI) crystallizes in the form of simple cubic structure. Determine the interatomic spacing \(d\) and the glancing angle at which an X-ray of wavelength of \(1.787\) A is reflected in the third order. The density of \(\mathrm{KCI}\) is \(1990 \mathrm{~kg} / \mathrm{m}^{3}\) and the molecular weight is \(74.6\).

What are Lauc equations for diffraction of \(X\) -rays by a crystalline solid? Show that the Bragg's equation is a special case of the Laue equations.

A bcc crystal is used to measure the wavelength of some X-rays. The Bragg angle for reflection from (110) plane is \(20.2^{\circ}\). What is the wavelength? The lattice parameter of the crystal is \(3.15 \mathrm{~A}\).
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